MinT - Best Known (32, 51, s)-Nets in Base 8

Best Known (32, 51, s)-Nets in Base 8

(32, 51, 256)-Net over F₈ — Constructive and digital

Digital (32, 51, 256)-net over F₈, using

3 times m-reduction [i] based on digital (32, 54, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 27, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(32, 51, 300)-Net in Base 8 — Constructive

(32, 51, 300)-net in base 8, using

3 times m-reduction [i] based on (32, 54, 300)-net in base 8, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
  - 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
    - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(32, 51, 455)-Net over F₈ — Digital

Digital (32, 51, 455)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵¹, 455, F₈, 19) (dual of [455, 404, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵¹, 511, F₈, 19) (dual of [511, 460, 20]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(32, 51, 61628)-Net in Base 8 — Upper bound on s

There is no (32, 51, 61629)-net in base 8, because

1 times m-reduction [i] would yield (32, 50, 61629)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1427 317607 012088 615975 243012 956124 912538 866564 > 8⁵⁰ [i]